Calculus- Early Transcendentals, 2021a

20 Review
The most familiar form of the equation of a straight line is:
y = mx + b.
Here m is the slope of the line: if you increase x by 1, the equation tells you that you have to increase
y by m; and if you increase x by Δx, theny increases by Δy = mΔx. The number b is called the y-
intercept, because it is where the line crosses the y-axis (when x = 0). If you know two points on a line,
the formula m =(y 2 − y 1 )/(x 2 − x 1 ) gives you the slope. Once you know a point and the slope, then the
y-intercept can be found by substituting the coordinates of either point in the equation: y 1 = mx 1 + b, i.e.,
b = y 1 − mx 1 . Alternatively, one can use the “point-slope” form of the equation of a straight line: start
with (y − y 1 )/(x − x 1 )=m and then multiply to get
(y − y 1 )=m(x − x 1 ),
the point-slope form. Of course, this may be further manipulated to get y = mx − mx 1 + y 1 ,whichis
essentially the “y = mx + b” form.
It is possible to find the equation of a line between two points directly from the relation m =(y −
y 1 )/(x − x 1 )=(y 2 − y 1 )/(x 2 − x 1 ), which says “the slope measured between the point (x 1 ,y 1 ) and the
point (x 2 ,y 2 ) is the same as the slope measured between the point (x 1 ,y 1 ) and any other point (x,y) on the
line.” For example, if we want to find the equation of the line joining our earlier points A(2,1) and B(3,3),
we can use this formula:
m = y − 1
x − 2 = 3 − 1 = 2, so that y − 1 = 2(x − 2), i.e., y = 2x − 3.
3 − 2
Of course, this is really just the point-slope formula, except that we are not computing m in a separate step.
We summarize the three common forms of writing a straight line below:
Slope-Intercept Form of a Straight Line
An equation of a line with slope m and y-intercept b is:
y = mx + b.
Point-Slope Form of a Straight Line
An equation of a line passing through (x 1 ,y 1 ) and having slope m is:
y − y 1 = m(x − x 1 ).
General Form of a Straight Line
Any line can be written in the form
Ax + By +C = 0,
where A,B,C are constants and A,B are not both 0.